Big name websites generally have enough users that email "just works". Smaller websites are more likely to use misguided measures such as a bad email validating regex (hello to anyone with a non-standard TLD!), only allowing gmail, or blacklisting domains like these.
From my personal experience it is best to have a secondary email account on a provider that is usually not blocked (like gmail), to keep your primary email account clean.
The only correct to validate email addresses is to just send a message there and see if the user can click the confirmation link.
Chances are that would be the next step in any signup flow anyway, so why introduce this artificial middle step of "validating the email address"?
Ever heard of Magento? They have that built in, at least in version 1. But it's a fixed list with "valid TLDs", anything not on that is not accepted when registering.
Feels strange, when you can't register on your own shop...
People can have more than one email address, so if your goal is "one account/offer/trial membership per real person", email ain't the way to achieve that, period.
Even worse are sites that disallow registering via "freemail providers" and require you to "use your ISPs or employer's". (Haven't seen this one in a while, but it definitely used to be a thing.)
My sites and apps have a blacklist and we don't allow email accounts from those. It's just me running this thing. If I had the security and engineering workforce of even a mid-sized tech company, I wouldn't have to do this. Alas.
I use owlmail.io for hundreds of accounts (major sites included) and haven't had an issue.